Exploring local and global stability of COVID-19 through numerical schemes

Respiratory sensitivity and pneumonia are possible outcomes of the coronavirus (COVID-19). Surface characteristics like temperature and sunshine affect how long the virus survives. This research article analyzes COVID-19 mathematical model behavior based on symptomatic and non-symptomatic individuals. In the reproductive model, the best result indicates the intensity of the epidemic. Our model remained stable at a certain point under controlled conditions after we evaluated a specific element. This approach is in place of traditional approaches such as Euler’s and Runge–Kutta’s. An unusual numerical approach known as the non-standard finite difference (NSFD) scheme is used in this article. This numerical approach gives us positivity. A dependable numerical analysis allowed us to evaluate different approaches and verify our theoretical results. Unlike the widely used Euler and RK4 approaches, we investigated the benefits of implementing NSFD schemes. By numerically simulating COVID-19 in a variety of scenarios, we demonstrated how our theoretical concepts work. The simulation findings support the usefulness of both approaches.


Parameter explanation of the model
The system which we discussed 43 consists of four differential equations.These equations represent the population N(t) , which is divided into four categories: susceptible S(t) , Infected I(t) , quarantined Q(t) , recovered R(t) where N(t) = S(t) + I(t) + Q(t) + R(t).
From the Fig. 1, we can derive a mathematical model for COVID-19.
where N is total population, so we reduced the N from the above equation for simplified the model.Hence our system now becomes, The flowchart of our model.

Parameters and their explanations
The following are some important concepts related to COVID-19:

Basic reproduction number (R 0 )
It's difficult to determine the exact number of secondary infections, as it depends on factors like the spread of the disease, population size, and duration of illness.However, epidemiological studies can provide a rough estimate called the basic reproduction number 44 .To calculate R 0 , we operate Model (2), which includes the translation matrix V(x) and the transmission matrix F(x).

Comparative analysis of schemes
In this article, our goal is to illustrate the superior performance of the NSFD scheme compared to other numerical schemes such as Euler Rk4.It has been noted that the Euler and Rk4 schemes exhibit divergence as the time step size increases.Conversely, the NSFD scheme maintains non-divergence even with considerably larger time step sizes.Consequently, we can assert that the NSFD scheme stands out as the optimal choice among numerical schemes.In the ensuing discussion, we will thoroughly examine all three schemes Euler, Rk4, and NSFD and provide both numerical data and graphical representations to showcase their respective performances.

The Euler scheme
The Euler scheme can be developed for the system (2) as shown below .
Vol:.( 1234567890) The RK-4 scheme Our challenges often use the RK-4 scheme unless otherwise noted.The RK-4 scheme was developed for system (2).wetake Vol.:(0123456789)The final stage is Where z 1 , z 2 , z 3 and z 4 are the weighted of The general form is.
Finally, we get

The NSFD scheme
The value of h (time step size) represents the numerical approximations of S(t),I(t),Q(t) , andR(t) at t = mh as per model (2).To denote these numerical estimates, we use the notation S n , I n , Q n , R n , where n is a non-negative integer 45,46 .Subsequently, model (2) enables us to express.

Therefore, we get
If 0 < W(0) < ϑ µ 3 , then by Gromwell's inequality is used to produce. Since This indicates that the feasible region becomes and the solutions of system (4) are bounded.
We evaluate the local stability of both equilibria in the NSFD scheme (4). Vol.:(0123456789)

Local stability of equilibria
Lemma 1 follows.We will apply the Schur-Cohn criterion 47,48 to demonstrate that the DFE point is LAS.

Lemma 1
The roots of L 2 − BL + C = 0 assurance L p < 1for p = 1,2 , ⇔ the necessities assumed in the succeeding remain satisfied.
anywhere B represents trace and C shows determinant of the Jacobian matrix.

Proof On
The information, we can express the Jacobian matrix in the following manner: In (5), f 1 , f 2 , f 3 , and f 4 (4) are provided.As you can see, every derivative used in ( 6) is noted here.
To explain the eigenvalues, we rely on the following assumptions: i.e.
Comparing Eq. ( 10) with The Schur-Cohn criterion is therefore satisfied whenever R 0 < 1 .As a result, NSFD scheme (4) DFE point Proof We develop it as follows, following a similar procedure to that by which we acquired the Jacobian matrix in Theorem 1 By placing DEE point E * , Eq. (11) To discuss the eigenvalues, we take i.e.

After simplification, (12) yields
It has two roots (13) < 1 , and To find other eigenvalues, we take ( 9) Accordingly, when R 0 > 1 , all of the longings of Schur-Cohn criterion quantified in Lemma 1 are pleased.So, on condition that that R 0 > 1 , the DEE point E * of the NSFD scheme (4) is LAS.

Global stability of equilibria
The process by which we obtained the Jacobian matrix in Theorem 1 is analogous to the one we use to derive it.
Proof Construct a Lyapunov function.
Using the inequality lnx ≤ x − 1 , (15) becomes The value of ( 16) can be expressed by making use of system (3).

Conclusions
In this study, we used a math model to analyze COVID-19, considering both symptomatic and asymptomatic conditions.We set a critical threshold value to explore the stability of key points in the continuous model.For this model, we developed algorithms like Euler, RK-4, and NSFD.Euler and RK-4's reliability depends on step size, with larger steps leading to more unpredictable results.In contrast, NSFD consistently converges regardless of step size.We examined the stability of key points for the NSFD scheme, considering both local and global aspects.By taking monotonic sequences into account, we assessed global stability.The NSFD scheme highlighted similarities between discrete and continuous models, offering advantages for society and medicine.Which we showed in the Figs. 2, 3 and 4.These findings can aid in predicting the course of the COVID-19 pandemic.Numerical simulations were incorporated at each step to support our theoretical framework.

ϑ 1 3 1 3 ,
Natural population growth: This refers to the increase in population due to births and immigration, minus the decrease in population caused by deaths and emigration β Transition rate from I to R: This refers to the rate at which infected individuals recover from COVID-19 and become immune d Cure rate for quarantined individuals: This is the rate at which individuals who are quarantined due to COVID-19 recover and become immune γ 3 COVID-19 infection-related death rate: This is the rate of deaths caused by COVID-19 among infected individuals α Rate of COVID-19 death among those under quarantine: This is the rate of deaths caused by COVID-19 among individuals who are under quarantine µ Natural population decline: This refers to the decrease in population due to deaths and emigration, minus the increase in population caused by births and immigration ψ Interval between infection and healing: This is the time it takes for an infected individual to recover from COVID-19 and become immune The equilibrium and the basic reproduction number ( R 0 ) Modelling the equilibrium state In order to find the disease-free equilibrium point (DFE), we can equate the model (2) to zero.It becomes easy to calculate DFE for model (2) when the DFE is represented by E 0 = ϑ µ 3 , 0,0, 0 .The given model (2) is instan- taneously cracked for the state-run variables S,I, Q, R, to invention the disease endemic equilibrium (DEE) point.If the DEE point is mentioned as E * (S * , I * , Q * , R * ) , then model (2) yields S * = (ϑ−βS * I * ) µ I * = βS * I * (µ3+γ3+α1+d) , Q * = dI * (µ3+ψ1+α1) , and R * = γ 3 I * +ψ 1 Q * µ 3